The Cointegration Analysis in Hypothesis of Heteroschedasticity: the Wild Bootstrap Test

STATISTICA, anno LXIII, n. 3, 2003

THE COINTEGRATION ANALYSIS IN HYPOTHESIS OF HETEROSCHEDASTICITY: THE WILD BOOTSTRAP TEST

M. Gerolimetto, I. Procidano

1. INTRODUCTION

The importance of the concept of cointegration (Engle, Granger, 1987) has in- creased in the recent years and it received much attention in the literature. Some of the seminal works are those of Johansen (1988, 1996), Phillips and Durlauf (1986), Phillips and Hansen (1990), Xiao and Phillips (1999). The cointegration analysis is usually carried out for macroeconomic and finan- cial time series that are often affected by heteroschedasticity. This aspect is ex- tremely relevant considering that homoschedasticity is one of the conditions un- derlying the applicability of the cointegration analysis procedures frequently used. So it is important to evaluate the robustness as regards heteroschedasticity of the procedures for the cointegration analysis and this is also the aim of this work. We consider and compare some traditional procedures and also an interesting proposal of a bootstrap test for cointegration. This test is originally a test for unit root (Procidano, Rigatti Luchini, 1999, 2000) which employs the bootstrap method called Wild Bootstrap. Most of the results about the estimators for non-stationary series are asymp- totic and their goodness has been evaluated by Montecarlo experiments, for example Schwert (1989), Diebold and Rudenbush (1991), Dejong et al. (1992). If the samples are finite, it seemed that the most frequently used tests exhibit serious level of size distortion and different powers. These results oriented the most recent researches about this theme to the de- velopment of new procedures of estimation and testing by using the bootstrap method. This is the context, where it is inserted the proposal of the Wild Boot- strap test for unit root, that we want here to extend to the cointegration analysis. In this work we want to compare, working by simulations, the robustness as regards heteroschedasticity of different procedures for cointegration analysis and, in particular, to verify if the Wild Bootstrap test is robust also in the context of the cointegration analysis. We consider the Dickey-Fuller test (DF: Dickey and Fuller, 1979), the Sargan-Bhargava test (DW: Sargan, Bhargava, 1983; Bhargava, 1986), the Johansen tests (Otrace and Omax: Johansen, 1988) and the Wild Bootstrap 604 M. Gerolimetto, I. Procidano test (B) in hypothesis of GARCH errors. This typology of heteroschedasticity has been chosen because it is considered general enough to represent the major part of situations of non-stationarity in variance, typical of several economical and fi- nancial time series. In this context, like in the unit root analysis (Procidano and Rigatti Luchini, 1999, 2000, 2001), the simulations show a robustness of the Wild Bootstrap test as regards heteroschedasticity, particularly for small samples. The article is organized as follows, section 2 introduces the notions of cointegration, section 3 explains the algorithm for the implementation of the wild Boot- strap Test, section 4 describes the simulation experiment, finally, section 5 con- cludes.

2. THE CONCEPT OF COINTEGRATION

The idea of a cointegration relationship between two or, more in general n>2, time series was intuitively proposed by Granger (1983) and then successively for- malized in the following definition: given two time series x1,t and x2,t , both I(1), without drift and trend, if it exists a linear combination

zxx EE (1) ttt121, 2 that is I(0), then x 1,t and x2,t are cointegrated and Ƣ =(Ƣ1,Ƣ2) is called vector the cointegration. Moreover, if x1,t and x2,t are both I(1) and cointegrated, they are always generated by the model ECM (Error Correction Mechanism) whose matricial formula- tion is the following: XX X tt Ƅƥ11 ơƢ()' tt 1 (2) where Xt=(x1,t, x2,t), ƥt=(ƥ1,t,ƥ2,t) is a white noise vector, Ƅ1 is a (2x2) matrix, Ƒ=ơƢ’ is a (2x2) matrix of rank 1, ơ=(ơ1,ơ2) is called speed of adjustment and Ƣ=(Ƣ1,Ƣ2) is the cointegration vector. Different procedures can be used to test for the existence of a cointegration relationship and they can be classified in two groups. The first group includes the “two steps procedures”: in the first step the cointegration relationship is estimated by a regression between the variables, in the second step the null hypothesis of unit root in the residuals is verified by the usual tests for unit root. Among the “two steps procedures” there are the Dickey-Fuller Test (DF), the Sargan- Bhargava Test (DW) and the Wild Bootstrap Test (B). More precisely, given the following model:

xxttt D 1 H t = 1,2,…T (3) where ƥt is a white noise with Ƴ2 variance, the statistic of the Dicey-Fuller test is: The cointegration analysis in hypothesis of heteroschedasticity: the wild bootstrap test 605

1/2 §·T ˆ 1 2 WD ( 1)S ¨¸ ¦xt 1 (4) ©¹t 1 1 T §TT ·§· 21 ˆ 2 ˆ 2 where ST (2)(¦ xxtt D 1 ) and D ¨¦¦xxtt11 ¸¨¸ x t t 1 ©tt 11 ¹©¹ The statistic of the Sargan-Bhargava test is:

TT 22 Rxx11 ¦¦()()tt xx t (5) tt 11 The second group includes the formulations that work directly on the time series, for example the Johansen test proposed in 1988. In this procedure the hypothesis of no cointegration is tested analysing the rank of the matrix Ƒ=ơƢ’ by maximum likelihood techniques. Johansen proposed two statistics:

n ˆ Otrace T ¦ log(1 Oi ) (6) ir 1 and

ˆ OOmax T log(1 r 1 ) (7)

ˆ ’ where Oi are the eigenvalues of the matrix Ƒ=ơƢ of the ECM model, T is the sample size and r, 0 d r

3. THE WILD BOOTSTRAP TEST

The B test proposed by Procidano and Rigatti Luchini (1999, 2000) to test for a unit root refers to the resample method proposed by Wu in 1986. This method is robust as regards heteroschedasticity and it consists of carrying out extractions from an external distribution with zero mean and unit variance whose choice represents a subjectivity element (Davidson and Flaichaire, 2000) that does not seem to significantly modify the results. The following is the general explanation of algorithm for the implementation of the B test: 606 M. Gerolimetto, I. Procidano

1 - We generate a time series xt from the model (3) in the null hypothesis ơ=1; 2 - We calculate the value of the ƴ statistics; 3 - We generate a replication of xt with the wild bootstrap and then we calculate the value of the ƴ statistics (indicated by ƴ*) on this replication; 4 - We repeat the step 3 J times: so we obtain the empirical distribution of ƴ*; 5 - We calculate the percentile corresponding the level of the test (1-Ƥ): we obtain so the extreme limit of the acceptance region with the wild bootstrap; 6 - We register if the statistics ƴ does not belong to the MacKinnon acceptance region or to the acceptance region calculated with the wild bootstrap following step 5; 7 - we repeat steps 1-2-3-4-5-6 N times: we obtain so the rejects proportion as regards the acceptance region calculated with the wild bootstrap.

When the B test is inserted in a context of cointegration analysis, the algorithm above described is applied to the residuals of the regression among the variables with the aim of establishing if the same residuals contain a unit root i.e. if the variables are cointegrated.

4. THE SIMULATION EXPERIMENT

In this section the performance of the B test is compared to that of the other cointegration tests. The comparison has been carried out by the simulation of 5000 time series of length T = 25, 50, 100, 200 for each of the two following schemes, partially referring to what Lee and Tse did in their work in 1996: 1) In the experiment for examining the size of the tests we generate non cointegrated systems with GARCH (1,1) errors. Let X t=(x1t,x2t, …,xNt)’ be an Nx1 vector of integrated series with X t ƥt (8)

The error vector ƥt=(e1t,e2t,…,eNt)’ is assumed to follow an N-variate conditional 2 2 normal distribution with E(eitɇFt-1)=0 and E(eit ɇFt-1)=Ƴit , where i = 1,2,…N, Ft-1 is the Ƴ-field generated by all the information available at time t-1 and

2 VIIIVit i0 iit 1,1e i ,2,1 it (9)

2) In the experiment for examinig the power of the tests we generate bivariate cointegrated systems with GARCH (1,1) errors. The system generated is

xxx1,t 0.2( 1, ttt 1 2, 1 ) H 1, (10) x 2,tt H 2, The cointegration analysis in hypothesis of heteroschedasticity: the wild bootstrap test 607

In the first case, the simulations permit to evaluate the size distortion of the different tests of cointegration expressed as proportion of rejects of the null hypothesis, in the second case the power of the different tests is estimated. The parameters of the GARCH model have been chosen like in Procidano and Rigatti Luchini (1999, 2000, 2001) i.e. considering quasi-integrated (ƶ1+ƶ2 # 1) and quasi-degenerete (ƶ0 # 0) structures.

5. RESULTS

In the following tables, for each choice of the GARCH parameters the size distortions of the tests are reported.

TABLE 5.1

Proportion of rejects (ƶ0=0.1 ƶң=0.3 ƶ2=0.6) Proportions of rejects out of 5000 tests, nominal level 0.95 500 replications, normal error distribution

Test T=25 T=50 T=100 T=200 ƶ0=0.1 ƶ1=0.3 ƶ2=0.6 DF 0.0668 0.0628 0.0621 0.0593 DW 0.0715 0.0678 0.0631 0.0618 OTrace 0.0701 0.0691 0.0591 0.0572 B 0.0598 0.0583 0.0569 0.0521 Omax 0.0737 0.0652 0.0636 0.0609

TABLE 5.2

Proportion of rejects (ƶ0=0.05 ƶң=0.3 ƶ2=0.65) Proportions of rejects out of 5000 tests, nominal level 0.95 500 replications, normal error distribution

Test T=25 T=50 T=100 T=200 ƶ0=0.05 ƶ1=0.3 ƶ2=0.65 DF 0.0684 0.0678 0.0659 0.0612 DW 0.0721 0.0682 0.0628 0.0624 OTrace 0.0798 0.0672 0.0622 0.0604 B 0.0619 0.0614 0.0593 0.0554 Omax 0.0816 0.0705 0.0624 0.0607

TABLE 5.3

Proportion of rejects (ƶ0=0.001 ƶң=0.3 ƶ2=0.699) Proportions of rejects out of 5000 tests, nominal level 0.95 500 replications, normal error distribution

Test T=25 T=50 T=100 T=200 ƶ0=0.001 ƶ1=0.3 ƶ2=0.699 DF 0.0692 0.0689 0.0678 0.0627 DW 0.0734 0.0712 0.0695 0.0653 OTrace 0.0786 0.0688 0.0646 0.0617 B 0.0563 0.0638 0.0621 0.0612 Omax 0.0833 0.0677 0.0636 0.0621 608 M. Gerolimetto, I. Procidano

From the analysis of tables 5.1, 5.2, 5.3, it is possible to see how the size distortion of all the tests tends to approach the nominal level as the sample size increases. Moreover, the size distortion exhibited by the B test is always the lowest for every sample size and for every parametric choice. This result is particularly important for small sample sizes (e.g. T = 25) where the test B permits to get proportions of rejects closer to the nominal level than those got by the other tests. As the sample size increases the difference becomes less evident because the size distortion tends to approach the nominal level. As far as the power is concerned, in the following tables the powers of the test are reported:

TABLE 5.4

Estimation of the power (ƶ0=0.1 ƶ=0.3 ƶ2=0.6) Estimation of the power of the tests, nominal level 0.95 500 replications, normal error distribution

Test T=25 T=50 T=100 T=200 ƶ0=0.01 ƶ=0.3 ƶ2=0.6 DF 0.175 0.492 0.628 0.915 DW 0.194 0.538 0.702 0.965 OTrace 0.190 0.412 0.850 0.926 B 0.201 0.498 0.711 0.971 Omax 0.219 0.505 0.916 0.992

TABLE 5.5

Estimation of the power (ƶ0=0.05 ƶ=0.3 ƶ2=0.65) Estimation of the power of the tests, nominal level 0.95 500 replications, normal error distribution

Test T=25 T=50 T=100 T=200

ƶ0=0.05 ƶң=0.3 ƶ2=0.65 DF 0.171 0.499 0.622 0.907 DW 0.168 0.532 0.697 0.968 OTrace 0.193 0.390 0.849 0.918 B 0.204 0.513 0.709 0.978 Omax 0.200 0.466 0.936 0.984

TABLE 5.6

Estimation of the power (ƶ0=0.001 ƶң=0.3 ƶ2=0.699) Estimation of the power of the tests, nominal level 0.95 500 replications, normal error distribution

Test T=25 T=50 T=100 T=200

ƶ0=0.001 ƶң=0.3 ƶ2=0.699 DF 0.179 0.502 0.620 0.909 DW 0.170 0.490 0.692 0.965 OTrace 0.184 0.489 0.859 0.952 B 0.191 0.507 0.698 0.970 Omax 0.190 0.517 0.927 0.989 The cointegration analysis in hypothesis of heteroschedasticity: the wild bootstrap test 609

Even though the power of the B test is not always the best, anyway it turns out to be high and compared to the DF test both compared to the other test in general. For every test the power increases as the sample size increases. These results seem to confirm the robustness of the Wild Bootstrap test with respect to heteroschedasticity also in the interesting context of the cointegration analysis. In our opinion, these are very important results, particularly because they could be profitably employed considering other heteroschedastic structures. Moreover, these results can be employed also in the recent and challenging context of fractional cointegration.

Dipartimento di Scienze Satistiche MARGHERITA GEROLIMETTO Università di Padova

Dipartimento di Statistica ISABELLA PROCIDANO Università Ca’ Foscari, Venezia

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RIASSUNTO

La verifica di cointegrazione in ipotesi di eteroschedasticità: il test bootstrap esterno In questo lavoro si vuole confrontare, tramite simulazioni, la robustezza rispetto al- l’eteroschedasticità di tipo GARCH (1,1) di alcune delle principali procedure proposte in letteratura per l’analisi di cointegrazione. In particolare, si considerano il test di Johansen e alcune procedure a due stadi, cioè il test Dickey-Fuller, il test Sargan-Bhargava ed un test bootstrap esterno. Quest’ultimo, confrontato con gli altri, ha fornito una buona performance specialmente alla bassa numerosità campionaria.

SUMMARY

The cointegration analysis in hypothesis of heteroschedasticity: the wild bootstrap test We consider the problem of comparing, by simulations, the robustness as regards heteroschedasticity GARCH (1,1) of some of the most important procedures proposed in literature for the cointegration analysis. In particular, we consider the Johansen test and some “two steps” procedures”, i.e. Dickey-Fuller test, Sargan-Bhargava test and an Ex- ternal Bootstrap test. The Bootstrap test performs very well, particularly for the lowest sample size.